Vector Spaces and Bases

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

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Stable and Supported Semantics in Continuous Vector Spaces

Proceedings of the Seventeenth International Conference on Principles of Knowledge Representation and Reasoning ◽

10.24963/kr.2020/7 ◽

2020 ◽

Author(s):

Yaniv Aspis ◽

Krysia Broda ◽

Alessandra Russo ◽

Jorge Lobo

Keyword(s):

Logic Program ◽

Matrix Multiplication ◽

Vector Spaces ◽

Continuous Space ◽

Continuous Vector ◽

Novel Approach ◽

Gradient Based ◽

Normal Logic ◽

Parameter Values ◽

Normal Logic Program

We introduce a novel approach for the computation of stable and supported models of normal logic programs in continuous vector spaces by a gradient-based search method. Specifically, the application of the immediate consequence operator of a program reduct can be computed in a vector space. To do this, Herbrand interpretations of a propositional program are embedded as 0-1 vectors in $\mathbb{R}^N$ and program reducts are represented as matrices in $\mathbb{R}^{N \times N}$. Using these representations we prove that the underlying semantics of a normal logic program is captured through matrix multiplication and a differentiable operation. As supported and stable models of a normal logic program can now be seen as fixed points in a continuous space, non-monotonic deduction can be performed using an optimisation process such as Newton's method. We report the results of several experiments using synthetically generated programs that demonstrate the feasibility of the approach and highlight how different parameter values can affect the behaviour of the system.

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ON SOFT FUZZY SOFT TOPOLOGICAL VECTOR SPACES AND DIFFERENTIATIONS

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.8.82 ◽

2020 ◽

Vol 9 (8) ◽

pp. 6145-6151

Author(s):

V. Visalakshi ◽

T. Yogalakshmi

Keyword(s):

Vector Spaces ◽

Topological Vector Spaces

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Risk Neutrality and Ordered Vector Spaces

SSRN Electronic Journal ◽

10.2139/ssrn.896201 ◽

2006 ◽

Cited By ~ 1

Author(s):

James C. Alexander ◽

Matthew J. Sobel

Keyword(s):

Vector Spaces ◽

Ordered Vector Spaces

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Finite dimensional locally convex cones

Filomat ◽

10.2298/fil1716111a ◽

2017 ◽

Vol 31 (16) ◽

pp. 5111-5116

Author(s):

Davood Ayaseha

Keyword(s):

Finite Dimension ◽

Convex Cones ◽

Vector Spaces ◽

Topological Vector Spaces ◽

Euclidean Topology ◽

Finite Dimensional ◽

Dimensional Cone ◽

Locally Convex Cones ◽

Locally Convex ◽

Special Case

We study the locally convex cones which have finite dimension. We introduce the Euclidean convex quasiuniform structure on a finite dimensional cone. In special case of finite dimensional locally convex topological vector spaces, the symmetric topology induced by the Euclidean convex quasiuniform structure reduces to the known concept of Euclidean topology. We prove that the dual of a finite dimensional cone endowed with the Euclidean convex quasiuniform structure is identical with it?s algebraic dual.

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Examples of theories of presheaf type

10.1093/oso/9780198758914.003.0011 ◽

2018 ◽

Author(s):

Olivia Caramello

Keyword(s):

Finite Groups ◽

Linear Independence ◽

Geometric Theory ◽

Vector Spaces ◽

Linear Orders ◽

Locally Finite ◽

Strong Unit ◽

Locally Finite Groups ◽

Simplicial Sets ◽

Separable Extensions

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.

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TransET: Knowledge Graph Embedding with Entity Types

Electronics ◽

10.3390/electronics10121407 ◽

2021 ◽

Vol 10 (12) ◽

pp. 1407

Author(s):

Peng Wang ◽

Jing Zhou ◽

Yuzhang Liu ◽

Xingchen Zhou

Keyword(s):

Link Prediction ◽

State Of The Art ◽

Score Function ◽

Graph Embedding ◽

Vector Spaces ◽

Knowledge Graph ◽

Semantic Features ◽

Knowledge Graphs ◽

Real World Datasets ◽

Low Dimensional

Knowledge graph embedding aims to embed entities and relations into low-dimensional vector spaces. Most existing methods only focus on triple facts in knowledge graphs. In addition, models based on translation or distance measurement cannot fully represent complex relations. As well-constructed prior knowledge, entity types can be employed to learn the representations of entities and relations. In this paper, we propose a novel knowledge graph embedding model named TransET, which takes advantage of entity types to learn more semantic features. More specifically, circle convolution based on the embeddings of entity and entity types is utilized to map head entity and tail entity to type-specific representations, then translation-based score function is used to learn the presentation triples. We evaluated our model on real-world datasets with two benchmark tasks of link prediction and triple classification. Experimental results demonstrate that it outperforms state-of-the-art models in most cases.

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Induced L-bornological vector spaces and L-Mackey convergence1

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-201599 ◽

2021 ◽

Vol 40 (1) ◽

pp. 1277-1285

Author(s):

Zhen-yu Jin ◽

Cong-hua Yan

Keyword(s):

Complete Lattice ◽

Vector Spaces ◽

Specific Description ◽

Equivalent Characterization ◽

Fuzzy Setting

Motivated by the concept of lattice-bornological vector spaces of J. Paseka, S. Solovyov and M. Stehlík, which extends bornological vector spaces to the fuzzy setting over a complete lattice, this paper continues to study the theory of L-bornological vector spaces. The specific description of L-bornological vector spaces is presented, some properties of Lowen functors between the category of bornological vector spaces and the category of L-bornological vector spaces are discussed. In addition, the notions and some properties of L-Mackey convergence and separation in L-bornological vector spaces are showed. The equivalent characterization of separation in L-bornological vector spaces in terms of L-Mackey convergence is obtained in particular.

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